Between meetings with the founder of Twitter and Square, who says cashless is nothing—he wants to get past plastic too.

Put it on Jack,” says the unassuming Internet CEO buying a couple of cappuccinos with no money in hand. After an awkward beat, the barista pecks at the iPad in front of her. “Oh, you’re on Square,” she says, and Jack Dorsey nods; that’s his company, the one he co-founded after Twitter, which enables merchants to take credit-card payments by phone plug-in. With Square’s new mobile app Card Case, users can pay participating merchants by simply saying their name, without ever having to swipe a card: When he walks up, Jack’s photo appears on the cashier’s screen, and after confirming his identity, she charges his credit card automatically. “Try it!” Dorsey, 35, told his million–plus Twitter followers last spring. “It feels like magic.”

Dorsey is handsome, like Carson Daly with a well-­proportioned head, and stylish in a reversible Barneys coat as we head out for a stroll on the High Line—he makes a point of aimlessly wandering for some time every day. He’s also a runner, waking up at 5:30 a.m. for meditation and a six-mile jog, takes his athletic shoes seriously, and has been reading up on barefoot running, though he’s skeptical of those freakish foot-glove things. “I don’t know if people want to see your toes in that way.”

If you believe the valuations—and you shouldn’t, entirely—Dorsey’s two start-ups, both still privately held, are worth more than $9 billion. But having Twitter to his name didn’t make Square easier, he says. “It was much harder the second time, actually, because everyone only saw what I did with Twitter. They said, ‘Why do you think you can work in the financial industry?’ That’s exactly why we’re going to do so well—the financial industry is built around blockers and people saying no.”

Dorsey founded Square after stepping down as Twitter’s CEO but returned to the social network in 2011 as executive chairman and chief product manager, meaning he’s now shuttling between two offices, running different companies, and remaining hands-on at both jobs. Just before New Year’s, he gave a tweeting tutorial to Rupert Murdoch. “He put his iPad down and said, ‘Jack, I want to get on this. How do I do it?’ ” Dorsey says. Since then, the scandal-drenched octogenarian has been blabbing away with typo-ridden abandon. “What’s been amazing is that he continues to use it every day. He doesn’t go through PR; he just fires them off.” In that way, it could be said, Murdoch is like Kanye West, who tweeted at two the night before about The Fancy, a “dope” new blogging site. Dorsey happens to sit on its board. “Everyone was very excited.”

Square’s clients are not really making waves (or facilitating uprisings) like Twitter’s, but Dorsey doesn’t seem to think that’s out of the question for a payment service that works via a smartphone’s headphone jack. “We weren’t doing any of that in two years with Twitter either,” Dorsey says. “Twitter really took off during the presidential election because the campaigns took to it. Here we are in 2012, and they’re taking to Square.” Dorsey’s right: Romney and Obama have recently adopted the app to take donations. And his pilot program to replace taxis’ backseat TVs with Square-enabled tablets was approved this past Thursday by the NYC Taxi and Limousine Commission. “New York doesn’t get enough credit, but it’s an early adopter of new technologies,” says Dorsey, who calls San Francisco, where he lives, “a little bit passive-aggressive.”

Dorsey was 18 when he moved east from St. Louis to take a job with a company whose website he’d hacked, and to study at New York University—from which he dropped out. “I should have gone to Columbia,” he says, looking out at the Hudson like it might carry him to campus. “But I didn’t want to be on the Upper West Side. I wanted to be downtown,” close to punk venues and Berkeley bands like Operation Ivy and Rancid. “I was hugely into punk rock, and I had a nose ring and dreadlocks, so I hung out on St. Marks Place.”

He keeps returning to that nose ring, like it’s the most important thing about him. “We raised money from Union Square Ventures, and I had the nose ring in,” he says of Twitter’s initial fund-raising. “No one ­really thought anything of it. But afterward I was going to Shake Shack—it was the summer, and I was wearing a full suit. I looked over, and there was another guy in a suit who had a nose ring. I took it out that night.

“It was a seamless ring, so literally you cannot take it out without tools,” Dorsey continues, though he’s now late for his next meeting in Soho. He got some grief for it from an H.R. lady while working as a freelancer pre-Twitter, programming the ticket system for boats to Alcatraz. “She said, ‘Put a Band-Aid on it if you want to come back to work here,’ or take it out, end of story. So I asked my boss if I should leave. He said, ‘No, just don’t go in the hallways ­anymore.’ ”